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Questions tagged [monomorphisms]

For questions related to monomorphisms, which are categorical generalizations of injective functions.

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Vakil's Foundations of Algebraic Geometry, Exercise 1.3.W - Pullback square exists if map is monic

In this exercise, the goal is to show that a morphism $\pi: X \to Y$ is a monomorphism iff the fibered product $X \times_Y X$ exists and that the induced diagonal morphism $\delta_\pi: X \to X \...
Corlio's user avatar
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Induced map in a cartesian square is a monomorphism

Consider the following cartesian square composed of four monomorphisms. The induced map from the pushout $Y\coprod Z$ to $T$ is also a monomorphism. Why is it the case? the pushout of a mono is a ...
Conjecture's user avatar
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In the category defined by $\le$ on $\mathbb{Z}$, every morphism defined is a monomorphism and an epimorphism

I am studying with Aluffi's "Algebra: Chapter 0", and got stuck on proving the statement on pg. 30 here: ... in Set, a function is an isomorphism if and only if it is both injective and ...
Nara's user avatar
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Categories in which there is a mono $A \to B$ iff there is an epi $B \to A$

Consider the property $P$ of a category $\mathcal{C}$ that for two objects $A$, $B$ in $\mathcal{C}$ there exists a monomorphism $A \to B$ iff there exists an epimorphism $B \to A$. Does the property $...
Smiley1000's user avatar
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About the universal morphism from the pushout of monomorphisms.

Let $A,B,C$ be objects in a category $\mathrm{C}$ and we have the pushout diagram of monomorphisms$\require{AMScd}$ \begin{CD} C @>>> A\\ @VVV @VVV\\ B @>>> A\bigsqcup\limits_C B\end{...
Epsilon's user avatar
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Show that if $G$ is the right adjoint of restriction-of-scalars along any ring homomorphisms $A\to B$, then $G$ is always a monomorphism

Let $A$ and $B$ be rings. Show that the right adjoint of restriction-of-scalars functor along any ring homomorphism $f: A \to B$ preserves injectives and show that the unit of this adjunction is ...
Squirrel-Power's user avatar
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Is a canonical morphism from the wedge sum to the product monic? A section?

Let $C$ be a category with a terminal object $\ast$. For two pointed objects, define their wedge sum $c\vee d$ as the pushout $$ \require{AMScd} \begin{CD} \ast @>>c_0> c \\ @VVd_0V@VV\...
Milten's user avatar
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1 answer
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Does the composition of a monic morphism with a terminal morphism make a monic morphism? [closed]

Consider $T$ to be a terminal element, $T \stackrel{f}{\rightarrowtail} A$ be a monic morphism (this can be shown by the terminal property of $T$) and $B\stackrel{\tau_B}{\rightarrow} T$ be the unique ...
Felipe Dilho's user avatar
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How do I prove the unicity of the terminal morphism obtained by being a equivalent monomorphism to one which has a terminal object as domain?

To clarify, consider in a category $\mathbf{C}$, a object $B$ and a terminal object $T$ both in Obj$_\mathbf{C}$, and a monomorphism $T \stackrel{f}{\rightarrowtail} B$. If $g$ in Mor$_\mathbf{C}$ is ...
Felipe Dilho's user avatar
3 votes
1 answer
113 views

Left exact functors preserve injective objects

Let $G$ be a group and let $\operatorname{\textbf{Mod}}_G$ be the abelian category of $G$-modules, which has enough injectives. Let $A\in \operatorname{Mod}_G$ and let $(-)^G$ the functor \begin{...
Conjecture's user avatar
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Mapping of a finite set onto itself

I'm studying Herstein's Topics in Algebra and stumbled upon the following problem (Chapter 0, section 2, problem 8): If the set S has a finite number of elements, prove the following: (a) If $\sigma$ ...
Alejandro Aguilar's user avatar
2 votes
2 answers
281 views

Monomorphism definition counterexample

I just cannot understand why definition of monomorphism is this: $$ \forall g_{1}, g_{2} : f \circ g_{1} = f \circ g_{2} \Rightarrow g_{1}=g_{2} $$ if we consider injection as a special case of ...
Kemsikov's user avatar
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Concrete category with a certain property involving monomorphisms

I am looking for a way to describe concrete categories with "good properties" in my work, and one of the properties I found that I want a concrete category $C$ to have is that whenever $g$ ...
Francisco's user avatar
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Define the image of a composition of maps

I just asked myself the following question and couldn't see if or where there is an issue. Let $f\colon A\to B$ an arrow and let $i_{1}\colon A_{1}\rightarrowtail A$ and $i_{2}\colon A_{2}\...
Dominique Fosse's user avatar
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Is $\phi$ a groupoid monomorphism?

Consider the binary algebraic structures $\langle\mathbb{R}, +\rangle$ and $\langle\mathbb{R}^{+}, \cdot\rangle$, and the mapping $\phi:\mathbb{R} \rightarrow\mathbb{R}^{+}$ where $\phi(x) = e^{x}$. ...
user avatar
5 votes
1 answer
96 views

Are monos in epis and epis in monos also monic or epic in the entire category?

Given any category $C$, let $\mathrm{Epi}(C)$ and $\mathrm{Mono}(C)$ denote the (generally non-full) subcategories of $C$ consisting of the epimorphisms and monomorphisms respectively. Then, is any ...
Geoffrey Trang's user avatar
1 vote
1 answer
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Why is the map from the kernel to the domain always a monomorphism?

I was reading about the universal property of kernels in category theory: if $f:A\to B$ is a morphism in a category with zero morphisms, the kernel of $f$ is an ordered pair $(K, k)$ where $K$ is an ...
node196884's user avatar
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2 answers
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Is there a name for a morphism which makes a left inverse act like a two-sided inverse?

$\newcommand{\Id}{\operatorname{Id}}$Consider a morphism $f : A \rightarrow B$ which has a left inverse $g$, i.e. $g \circ f = \Id_A$. (That is, $f$ is a split monomorphism.) Of course, we don't ...
Sambo's user avatar
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2 votes
1 answer
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Carrying out a proof in an Abelian category without using an embedding theorem

Let $C$ be an abelian category, and consider the following diagram $\require{AMScd}$ \begin{CD} \ker(C\to B)@>h>> C@>>f\circ p>B \\ @. @VpVV @V idVV \\ \ker(A\to B) @>k>>A @&...
t_kln's user avatar
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2 votes
1 answer
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Monomorphisms induce monomorphisms between cokernels

All of the following are discussed in an abelian category $\mathcal{C}$. Given $X \stackrel{g}{\rightarrow} Y \stackrel{f}{\hookrightarrow}Z$. Consider the morphism $$\mathrm{coker}\ g \stackrel{k}{\...
Chen's user avatar
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If $f:Y \to X$ is a mono in $\textsf{Set}^{\mathbb{C}^{op}}$, then $f$ factors uniquely through $Y \xrightarrow{g} A \stackrel{i}{\hookrightarrow} X$.

I have taken an introductory course in category theory and would like to learn more about presheaves. Currently I am working through "Generic figures and their glueings" by Marie La Palme ...
user11718766's user avatar
1 vote
1 answer
29 views

Does subclass of morphisms closed under wide pullbacks necessarily consist of only monomorphisms?

For category $\mathcal{C}$, does subclass $M \subseteq \operatorname{Mor}(\mathcal{C})$ closed under even largely wide pullbacks necessarily consist of only monomorphisms? The assumption being more ...
PPP's user avatar
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1 answer
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If for monos $u\leq v$ and $v\leq u$ then their domains are isomorphic

I'm unable to prove that if a mono $u:B\to A$ is less then mono $v:C\to A$ by $f:B\to C$ with $v\circ f=u$ and also $v\leq u$ by $u\circ g=v$ then $B\cong C$ by using some morphisms as above, but I ...
user122424's user avatar
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4 votes
5 answers
277 views

Category where not every mono is regular

I'm looking for a category with the titular property. Since $f : A \to B$ is supposed to be mono, the desired factorisation is automatically unique (correct?). Thus, I would need to find a category in ...
Jos van Nieuwman's user avatar
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1 answer
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Is every split mono regular?

Let $f : A \to B$ be a split mono: $(\exists g : B \to A)(g \circ f = \text{id}_A)$. (It it clear that $f$ is in particular a monomorphim.) I want to show that it is regular, i.e., there is an object $...
Jos van Nieuwman's user avatar
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1 answer
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map from equaliser/kernel is a monomorphism

I'm going through lecture notes on categories and homological algebra. There seems to be a very basic fact that's eluding me. Let $C$ be a category and $f,g\in Hom(X_0,X_1)$, such that $K=\ker(f,g)$ ...
t_kln's user avatar
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2 votes
1 answer
406 views

Monomorphisms and epimorphisms in abelian categories

Let $\mathsf{C}$ be a an abelian category. Let $f \colon X \rightarrow Y$ be a morphism in $\mathsf{C}$. I am looking at the following two statements: The morphism $f$ is a monomorphism if and only ...
Margaret's user avatar
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2 answers
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any $\lambda-$pure morphism in a $\lambda$-accessible category is a monomorphism

I'm trying to understand the proof of the proposition 2.29 in the book Locally presentable and accessible categories which is also given in the snippet below. I've got stuck in the -2nd paragraph ...
user122424's user avatar
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1 vote
0 answers
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subobject in Set

I cannot compute the coequaliser of $s$ and $id_M$ in $\mathbf{Set}$: My problem is that if $s$ switches $a$ and $b$ in $M=\{a,b\}$ how will then look the coequaliser $M \rightrightarrows M \to U$ ...
user122424's user avatar
  • 3,978
4 votes
2 answers
100 views

Monomorphisms in the category of finite-dimensional Lie algebras injective?

Motivation: I'm trying to figure out whether the image of an injective Lie group homomorphism (between simply connected Lie groups) under the Lie functor is injective. Since the Lie functor is an ...
kringelton4000's user avatar
1 vote
0 answers
100 views

Compatibility of pullbacks with an equivalence relation

I'm currently working on the proof of the existence of the sheafification in Notes on Grothendieck topologies, fibered categories and descent theory , but i currently stuck on a statement in the proof ...
Muster Maxfrau's user avatar
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0 answers
84 views

When $g \circ f$ is an extremal monomorphism, $f$ must be an extremal monomorphism. Why?

I couldn't prove part (2) of the proposition 7.62 in Adameck's book (Abstract and Concrete Categories: The Joy of Cats). It states that if $g \circ f$ is assumed to be an extremal monomorphism, then ...
Ali Koohpaee's user avatar
1 vote
1 answer
144 views

Why every regular monomorphism is a monomorphism?

I read in section 7.57 of Adameck's book (Abstract and Concrete Categories The Joy of Cats) that due to the uniqueness requirement in the definition of equalizer, every regular monomorphism must be a ...
Ali Koohpaee's user avatar
1 vote
1 answer
65 views

Example of non-balanced category.

I have a claim that in the category of torsion-subgroup-free abelian groups any nonzero homomorphism from $\mathbb{Z}$ to $\mathbb{Z}$ is mono and epi. I am struggling to prove that it is indeed the ...
Michal Dvořák's user avatar
1 vote
1 answer
111 views

Interchange map in simplicial sets is a monomorphism?

In the category of simplicial sets there is an 'interchange' map $$h : (A \times B) * (C \times D) \to (A * C) \times (B * D)$$ given by $$h = (\pi_A * \pi_C, \pi_B * \pi_D)$$ where $\times$ is the ...
nasosev's user avatar
  • 469
5 votes
2 answers
281 views

Injective map of schemes that is not a monomorphism

I know that it is not true that for a map $f \colon X \rightarrow Y$ of schemes, injectivity (on underlying sets) of $f$ gives a monomorphism in the category of schemes. Stronger assumptions are ...
Maximilian Keßler's user avatar
1 vote
1 answer
35 views

Monomorphism between ${\rm GL}(1,4)$ and ${\rm GL}(2,2)$ that corresponds to the inclusion of $A_3$ in $S_3$

I am currently solving as many exercises as possible from General linear groups and special linear groups because I am not good in this part of algebra. The question here is to construct one such ...
philips's user avatar
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5 votes
1 answer
60 views

Are monomorphisms in the category of Artinian rings injective?

The argument for the category of all rings works just as well for the category of Noetherian rings, since $\mathbb{Z}[x]$ is Noetherian. However, $\mathbb{Z}[x]$ is not Artinian. So, is it still true ...
Geoffrey Trang's user avatar
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0 answers
145 views

Monomorphism and mapping of vector spaces

Let $V$ and $W$ be two vector spaces, and let $f:V \rightarrow W$ be a linear mapping. Show that $f$ is a Monomorphism only when $f$ maps linearly independent sets in $V$ to linearly independent sets ...
adr555's user avatar
  • 77
1 vote
1 answer
60 views

Morphism between free abelian groups is 1-1; Vick Prop. 1.9

Proposition 1.6 in Vick's Homology Theory states: If $X$ is a space and $\{X_{\alpha}:\alpha \in A\}$ are the path components of $X$, then $$H_{k}(X) \approx \sum_{\alpha \in A}H_{k}(X_{\alpha}).$$ ...
Jorge Zazueta's user avatar
1 vote
2 answers
476 views

Why doesn't the localization map $A \to A_f$ show that open embeddings of schemes are not always monomorphisms?

Suppose $\iota: X \hookrightarrow Y$ is an open embedding of schemes. We can assume $X$ is an open subscheme of $Y$. On the level of sets, $\iota$ is injective. I wish to see that the pullback map $\...
user5826's user avatar
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0 votes
2 answers
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In general, how does one show that a mapping between two groups is a monomorphism? Example included.

The definition of a group monomorphism is as follows: Let $G$ and $H$ be groups and suppose $\phi:G\to H$ is a group homomorphism. Then $\phi$ is a group monomorphism if and only if $\phi$ is an ...
MrStormy83's user avatar
1 vote
2 answers
56 views

Doubt proving that $(\forall f,f' \quad g\circ f = g\circ f' \implies f=f' ) \implies g \text{ is injective}$?

I am trying to understand the proof of: (ii) The function $g : b \to c$ is mono iff the following condition holds: for any two functions $f,f' : a \to b$, $g \circ f = g \circ f'$ implies $f=f'$ ...
Red Banana's user avatar
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2 votes
0 answers
63 views

Categories with colimit-stable monomorphisms

In Definition 2.12 of https://arxiv.org/pdf/1409.3805.pdf, Adamek defines the notion of a cocomplete category having stable monomorphisms, meaning that small coproducts of monomorphisms are ...
User7819's user avatar
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1 vote
1 answer
180 views

$E(M)$ is indecomposible if and only if $M$ is uniform.

I tried to prove the above mentioned statement written in the book "Serial Rings" by Gennadi Puninski. For the first part I assumed that the module $E(M)$ is indecomposable. I was going by ...
Potato's user avatar
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2 votes
0 answers
103 views

Do open continuous maps/local homeomorphisms between locales possess adjoints?

Recently I started learning "theory of locales" (point-free topology) by my-self. While being a very beautiful, natural subject and parallel to point-set topology, some of its notions are ...
Bumblebee's user avatar
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2 votes
1 answer
95 views

Confusing Exercise in Hilton/Stammbach's *A Course in Homological Algebra*

An exercise in Hilton/Stammbach's book begins: "Show that $j : X_0 \to X$ in $\mathfrak{T}$ [the category of topological spaces and continuous maps] is a homeomorphism of $X_0$ into $X$ if and ...
Nick A.'s user avatar
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3 votes
0 answers
69 views

correspondence between epi and monos

I want to know if there is a problem with this argument: Let $C$ be a category and $F$ be a faithful functor $C^{op}\rightarrow C$, with left adjoint $F^{op}: C\rightarrow C^{op}$ If $f: A\rightarrow ...
Sajad's user avatar
  • 121
3 votes
1 answer
153 views

Subobject classifier of $C$ is the same as the terminal object of $D$

Let $C$ be a category, then construct the category $D$ whose objects are monomorphisma of $C$ and arrows are pullback squares of these monics. Show that the subobject classifier of $C$ is the same as ...
user850424's user avatar
2 votes
1 answer
189 views

A monomorphism from a colimit

Let $D$ be an upward directed poset, and suppose I have a diagram $F: D\to C$ in a cocomplete category such that $F(d)\to c$ is a monomorphism for all $d\in D$ for some $c\in C.$ Is it true that $\...
Bumblebee's user avatar
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